Semiclassical Trace Formula for the Bochner–Schrödinger Operator

We study the semiclassical Bochner–Schrödinger operator \(H_{p}=\frac{1}{p^2}\Delta^{L^p\otimes E}+V\) on tensor powers \(L^p\) of a Hermitian line bundle \(L\) twisted by a Hermitian vector bundle \(E\) on a Riemannian manifold of bounded geometry. For any function \(\varphi\in C^\infty_c(\mathbb R)\), we consider the bounded linear operator \(\varphi(H_p)\) in \(L^2(X,L^p\otimes E)\) defined by the spectral theorem. We prove that its smooth Schwartz kernel on the diagonal admits a complete asymptotic expansion in powers of \(p^{-1}\) in the semiclassical limit \(p\to \infty\). In particular, when the manifold is compact, we get a complete asymptotic expansion for the trace of \(\varphi(H_p)\).

DOI 10.1134/S1061920825600333